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Golay Complementary Sequences of Arbitrary Length and Asymptotic Existence of Hadamard Matrices (2401.15381v1)

Published 27 Jan 2024 in cs.IT and math.IT

Abstract: In this work, we construct $4$-phase Golay complementary sequence (GCS) set of cardinality $2{3+\lceil \log_2 r \rceil}$ with arbitrary sequence length $n$, where the $10{13}$-base expansion of $n$ has $r$ nonzero digits. Specifically, the GCS octets (eight sequences) cover all the lengths no greater than $10{13}$. Besides, based on the representation theory of signed symmetric group, we construct Hadamard matrices from some special GCS to improve their asymptotic existence: there exist Hadamard matrices of order $2t m$ for any odd number $m$, where $t = 6\lfloor \frac{1}{40}\log_{2}m\rfloor + 10$.

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